Integral representation and Γ -convergence of variational integrals with p ( x ) -growth

Alessandra Coscia; Domenico Mucci

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • Volume: 7, page 495-519
  • ISSN: 1292-8119

Abstract

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We study the integral representation properties of limits of sequences of integral functionals like ∫ f ( x , D u ) d x under nonstandard growth conditions of ( p , q ) -type: namely, we assume that | z | p ( x ) ≤ f ( x , z ) ≤ L ( 1 + | z | p ( x ) ) . Under weak assumptions on the continuous function p ( x ) , we prove Γ -convergence to integral functionals of the same type. We also analyse the case of integrands f ( x , u , D u ) depending explicitly on u ; finally we weaken the assumption allowing p ( x ) to be discontinuous on nice sets.

How to cite

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Coscia, Alessandra, and Mucci, Domenico. "Integral representation and $\sf \Gamma $-convergence of variational integrals with ${p(x)}$-growth." ESAIM: Control, Optimisation and Calculus of Variations 7 (2002): 495-519. <http://eudml.org/doc/245521>.

@article{Coscia2002,
abstract = {We study the integral representation properties of limits of sequences of integral functionals like $\int f(x,Du)\,\{\rm d\}x$ under nonstandard growth conditions of $(p,q)$-type: namely, we assume that\[ \vert z\vert ^\{p(x)\}\le f(x,z)\le L(1+\vert z\vert ^\{p(x)\})\,. \]Under weak assumptions on the continuous function $p(x)$, we prove $\Gamma $-convergence to integral functionals of the same type. We also analyse the case of integrands $f(x,u,Du)$ depending explicitly on $u$; finally we weaken the assumption allowing $p(x)$ to be discontinuous on nice sets.},
author = {Coscia, Alessandra, Mucci, Domenico},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {integral representation; $\Gamma $-convergence; nonstandard growth conditions; Gamma-convergence; integral functionals},
language = {eng},
pages = {495-519},
publisher = {EDP-Sciences},
title = {Integral representation and $\sf \Gamma $-convergence of variational integrals with $\{p(x)\}$-growth},
url = {http://eudml.org/doc/245521},
volume = {7},
year = {2002},
}

TY - JOUR
AU - Coscia, Alessandra
AU - Mucci, Domenico
TI - Integral representation and $\sf \Gamma $-convergence of variational integrals with ${p(x)}$-growth
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 7
SP - 495
EP - 519
AB - We study the integral representation properties of limits of sequences of integral functionals like $\int f(x,Du)\,{\rm d}x$ under nonstandard growth conditions of $(p,q)$-type: namely, we assume that\[ \vert z\vert ^{p(x)}\le f(x,z)\le L(1+\vert z\vert ^{p(x)})\,. \]Under weak assumptions on the continuous function $p(x)$, we prove $\Gamma $-convergence to integral functionals of the same type. We also analyse the case of integrands $f(x,u,Du)$ depending explicitly on $u$; finally we weaken the assumption allowing $p(x)$ to be discontinuous on nice sets.
LA - eng
KW - integral representation; $\Gamma $-convergence; nonstandard growth conditions; Gamma-convergence; integral functionals
UR - http://eudml.org/doc/245521
ER -

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